# Towards Accurate Guided Diffusion Sampling through Symplectic Adjoint Method

arXiv: https://arxiv.org/abs/2312.12030 · HTML: https://mlsanity.com/paper/2312.12030/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 33 of its 38 references that are in the map.
- First posted on arXiv 19 Dec 2023.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 33)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Adding Conditional Control to Text-to-Image Diffusion Models](https://mlsanity.com/paper/2302.05543/) (2023)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [DreamBooth: Fine Tuning Text-to-Image Diffusion Models for Subject-Driven Generation](https://mlsanity.com/paper/2208.12242/) (2022)
- [An Image is Worth One Word: Personalizing Text-to-Image Generation using Textual Inversion](https://mlsanity.com/paper/2208.01618/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Imagen Video: High Definition Video Generation with Diffusion Models](https://mlsanity.com/paper/2210.02303/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Palette: Image-to-Image Diffusion Models](https://mlsanity.com/paper/2111.05826/) (2021)
- [Pick-a-Pic: An Open Dataset of User Preferences for Text-to-Image Generation](https://mlsanity.com/paper/2305.01569/) (2023)
- [Tune-A-Video: One-Shot Tuning of Image Diffusion Models for Text-to-Video Generation](https://mlsanity.com/paper/2212.11565/) (2022)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [AudioLDM: Text-to-Audio Generation with Latent Diffusion Models](https://mlsanity.com/paper/2301.12503/) (2023)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [Zero-shot Image-to-Image Translation](https://mlsanity.com/paper/2302.03027/) (2023)
- [MagicVideo: Efficient Video Generation With Latent Diffusion Models](https://mlsanity.com/paper/2211.11018/) (2022)
- [Universal Guidance for Diffusion Models](https://mlsanity.com/paper/2302.07121/) (2023)
- [FreeDoM: Training-Free Energy-Guided Conditional Diffusion Model](https://mlsanity.com/paper/2303.09833/) (2023)
- [EDICT: Exact Diffusion Inversion via Coupled Transformations](https://mlsanity.com/paper/2211.12446/) (2022)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Dreamix: Video Diffusion Models are General Video Editors](https://mlsanity.com/paper/2302.01329/) (2023)
- [End-to-End Diffusion Latent Optimization Improves Classifier Guidance](https://mlsanity.com/paper/2303.13703/) (2023)
- [MagicEdit: High-Fidelity and Temporally Coherent Video Editing](https://mlsanity.com/paper/2308.14749/) (2023)

## Built on by (most cited first, 8 of 8)

- [Controllable Generation with Text-to-Image Diffusion Models: A Survey](https://mlsanity.com/paper/2403.04279/) (2024)
- [Gradient Guidance for Diffusion Models: An Optimization Perspective](https://mlsanity.com/paper/2404.14743/) (2024)
- [Gradient-Free Generation for Hard-Constrained Systems](https://mlsanity.com/paper/2412.01786/) (2024)
- [A Survey on Personalized Content Synthesis with Diffusion Models](https://mlsanity.com/paper/2405.05538/) (2024)
- [E-Motion: Future Motion Simulation via Event Sequence Diffusion](https://mlsanity.com/paper/2410.08649/) (2024)
- [Efficiently Access Diffusion Fisher: Within the Outer Product Span Space](https://mlsanity.com/paper/2505.23264/) (2025)
- [Greed is Good: A Unifying Perspective on Guided Generation](https://mlsanity.com/paper/2502.08006/) (2025)
- [Multi-Step Guided Diffusion for Image Restoration on Edge Devices: Toward Lightweight Perception in Embodied AI](https://mlsanity.com/paper/2506.07286/) (2025)

## Abstract (arXiv)

Training-free guided sampling in diffusion models leverages off-the-shelf pre-trained networks, such as an aesthetic evaluation model, to guide the generation process. Current training-free guided sampling algorithms obtain the guidance energy function based on a one-step estimate of the clean image. However, since the off-the-shelf pre-trained networks are trained on clean images, the one-step estimation procedure of the clean image may be inaccurate, especially in the early stages of the generation process in diffusion models. This causes the guidance in the early time steps to be inaccurate. To overcome this problem, we propose Symplectic Adjoint Guidance (SAG), which calculates the gradient guidance in two inner stages. Firstly, SAG estimates the clean image via n function calls, where n serves as a flexible hyperparameter that can be tailored to meet specific image quality requirements. Secondly, SAG uses the symplectic adjoint method to obtain the gradients accurately and efficiently in terms of the memory requirements. Extensive experiments demonstrate that SAG generates images with higher qualities compared to the baselines in both guided image and video generation tasks.